Eigenchannel analysis of super-resolution far-field sensing with a randomly scattering analyzer
The paper presents a method of analysis for randomly scattering analyzers that offer far-subwavelength
spatial resolution with coherent light. It discusses the eigenvalue distributions of the transmission matrix
in scattering systems and explores the impact of reducing the angular support of the set of plane waves
used in transmission eigenchannel analysis. The paper also discusses the statistics and properties of the
T matrix and its potential for super-resolution far-field sensing. It concludes by discussing the limitations
and potential applications of randomly scattering analyzers.
Here are some elaborations on the issues discussed in the paper:
1. Eigenvalue Distributions: The paper explores the eigenvalue distributions of the transmission matrix
(T matrix) in scattering systems. It mentions the quarter-circle distribution, which has been observed
experimentally in the singular value distribution of transmission matrices for disordered media [2].
However, the paper argues that assumptions about the T matrix having a quarter-circle eigenvalue
distribution are incompatible with the properties of the matrix [2].
2. Impact of Angular Support: The paper investigates the effect of reducing the angular support of the
set of plane waves used in the T matrix. It observes that reducing this set of plane waves has a similar
effect on the eigenvalue distribution as reducing the amount of scatter in the analyzer. Specifically, the
peak near zero in the eigenvalue distribution shrinks in height [3].
3. Bimodal Distribution: The paper introduces the bimodal distribution and the quarter-circle
distribution as theoretical eigenvalue distributions from random matrix theory. It compares these
distributions to the empirical eigenvalue distributions derived from numerical simulations.
Normalizations are performed to make the comparisons appropriate [5].
4. Potential Applications: The paper discusses the potential applications of randomly scattering
analyzers. It mentions the possibility of using asymmetric-transfer-function metasurfaces to design
scattering analyzers with increased sensitivity compared to random analyzers. It also suggests optimizing
the spatial profile of the incident field to improve precision in parameter estimations through diffuse
media. The paper highlights the potential for super-resolution sensing and microscopy where farsubwavelength spatial information is important [8].
These elaborations provide a brief overview of the issues discussed in the paper. Please refer to the
corresponding citations for more detailed information.
## Introduction
- The paper discusses the concept of super-resolution spatial sensitivity from changes in the position of
an object with a structured coherent background field.
- The sensitivity relates to the properties of the random analyzer and offers substantial impact in a
variety of applications.
## Eigenchannel Analysis
- A method of analysis for a randomly scattering analyzer offering far-subwavelength spatial resolution
with coherent light is presented.
- Enhanced spatial resolution is shown to be possible because of relative motion with a structured field
and the resulting information available.
- Detected information through a scattering analyzer results in enhanced spatial sensitivity with motion
of an object in a structured field.
## Random Analyzer
- A random analyzer, in principle, allows subwavelength sensitivity whose resolution is limited only by
measurement accuracy and precision.
- Use of a random analyzer offers substantial impact in a variety of applications.
## Disordered Media
- The physics of disordered media is of substantial importance in quantum transport and statistical
optics.
- Despite the information that, in principle, exists in heavily scattered coherent waves, extraction or
control remains challenging.
## Random Matrix Theory
- The paper explores the topic of randomly scattering analyzers using random matrix theory.
- The ability of such an analyzer to enhance sensitivity is accompanied by certain changes in the
probability distribution of the eigenvalues of the transmission matrix that models the analyzer.
## Correlations and Eigenvalues
- The ensemble-averaged intensity correlation is written in terms of the correlation of the detected
fields through a polarizer at different object positions.
- The scattering analyzer can be understood through a scattering matrix and, in particular, the
transmission submatrix T.
## Finite Element Method Simulations
- The finite element method simulations demonstrate the relationship between the amount of scatter in
the analyzer and the eigenvalue distribution.
- These results are examined by comparing them to existing distributions from the literature.
## Conclusion
- The paper provides a deeper understanding of the underlying mechanisms that result in enhanced
sensitivity.
- Such statistics are merely sufficient, but not necessary, for achieving subwavelength far-field
sensitivity.
## Introduction
- The scattering matrix S relates a set of incident and scattered modes.
- A plane-wave basis for generating a transmission matrix is relevant for applications.
## Normalized field correlation
- The normalized field correlation can be written using the generalized Wiener-Khinchin theorem.
- The normalized field is different from the normalized intensity due to the relationship between the
two.
- The correlation involves a sum over plane waves with randomly distributed complex coefficients.
## Discretization of transmission matrix
- The plane-wave transmission matrix describing the analyzer is discretized.
- The eigenvalue decomposition of the transmission matrix is analyzed using tools from random matrix
theory.
- The average normalized discrete intensity correlation in the detector half space is obtained.
## FEM simulation
- The specific nature of the eigenvalues and the character of the random elements of T dictate the
characteristics of the analyzer.
- Numerical simulations give a solution to the sinusoidal steady state Maxwell’s equations on a mesh.
- The FEM solution domain uses a scattered field formulation and perfectly matched layers on the
incident and transmission sides.
- The transverse wave numbers of the incident plane waves are assigned values based on the periodic
geometry.
- The empirical density functions of the normalized transmission eigenvalues are compared to the
theoretical bimodal probability distribution.
## Conclusion
- The study of eigenchannel analysis of super-resolution imaging using random scattering analyzers is
important.
- The results of the FEM simulations provide insight into the characteristics of the analyzer.
- Further research is needed to fully understand the behavior of the eigenvalues near T=1.
## Simulation Setup
- The analyzer is discretized into square regions of 200 nm ×200 nm, each of which is randomly assigned
material (dielectric or free space) at a fill factor of 50%.
- The detector plane is a distance of 4 λ from the analyzer and spans the entire breadth of the geometry
( Lx).
## Eigenvalue Distributions
- Two theoretical eigenvalue distributions from random matrix theory are introduced: the bimodal
distribution and the quarter-circle distribution.
- Normalized eigenvalues are derived from the theoretical distributions and compared to the empirical
eigenvalue distributions derived from numerical simulations.
- The bimodal density function for Tn is given by pbm(T)=T0/T√1−T, for T∈[δ,1], where δ is positive and
small compared to unity.
- The quarter-circle distribution is used to compare the singular values of both Re(T) and Im(T).
## Estimated Mean-Free Path Length
- The Beer-Lambert law is used to estimate the mean-free path length of the scattering material with
/epsilon1r=5.
- The fitted curve gives an estimated value of ˆ/lscript≈2.8µm.
## Bimodal Distribution
- The bimodal distribution has arisen when discussing the transmission matrix eigenvalues in scattering
systems.
- The bimodal density function for Tn is given by pbm(T)=T0/T√1−T, for T∈[δ,1], where δ is positive and
small compared to unity.
- The correct choice of δ is discussed in Sec. IV C 4.
## Quarter-Circle Distribution
- The quarter-circle distribution is used to compare the singular values of both Re(T) and Im(T).
- Each set of singular values is normalized using methods analogous to (5).
## Normalization
- Normalized eigenvalues are derived from the theoretical distributions and compared to the empirical
eigenvalue distributions derived from numerical simulations.
- All eigenvalues from different slab types are normalized with the same scaling value to compare all of
these eigenvalue distributions to one another.
## Empirical Data
- The calculated density functions of the normalized eigenvalues (of THT) are given in Fig. 2 for each level
of scatter.
- The last column of Table I contains data that describes how closely the detected speckle fields satisfy
the assumption of zero-mean circular Gaussian statistics.
## Transmission Coefficient
- The transmission coefficient t is calculated for each of the ten instances of the medium-, high-, and
highest-scatter configurations.
- The exponential model in (6) is fit to the data, and the fitted mean-free path is ˆ/lscript≈2.8 µm.
## Empirical eigenvalue distributions
- Figure 2 shows empirical density functions for the normalized transmission eigenvalues T of the matrix
THT for a plane-wave spectrum with N=141.
- Increasing analyzer scatter representative of the diffusion regime is expected to yield a bimodal
density, a trend supported by normalized eigenvalues with a space-based T.
- No clear pattern is evident in Fig. 2 for the relatively large eigenvalues (close to 1) and the amount of
scatter considered.
## Statistics of the bimodal distribution
- The bottom and top 10% means of the normalized transmission eigenvalues are shown in Table II.
- Figure 2 clearly shows an increasing proportion of closed channels as the analyzer scatter increases,
and the bottom 10% means in Table II agree with this trend.
- The top 10% means also demonstrate an increasing proportion of transmissive channels as the scatter
increases, though no clear corresponding trend appears in the figure.
## Impact of angular support
- Reducing the angular support of the set of plane waves used in T has a somewhat similar effect on the
eigenvalue distribution as reducing the amount of scatter in the analyzer.
- No trend appears near T=1.
## Assumptions of the bimodal distribution
- For further insight into potential reasons for the absence of the high-eigenvalue peak in Fig. 2, we turn
to the derivation of the bimodal distribution.
- It has been noted that, for the scattering regime where the bimodal distribution is an appropriate
model for the transmission eigenvalues, the probability distribution for x tends toward uniform.
- The resulting xn are compared to a uniform distribution in Fig. 5.
- Figure 5 shows that these xn are not, in fact, uniformly distributed, meaning that the scattering regime
resulting from our simulations is not the same as the one in which we would expect to see a bimodal
eigenvalue distribution.
- This explains the absence of the high-eigenvalue peak in Fig. 2.
## Relationship between two figures
- Equation (9) is strictly monotonically decreasing for T∈(0,1), which is the range of T plotted in Fig. 2.
- Therefore, the left side of Fig. 5 is related to the right side of Fig. 2(a), and vice versa.
## Expectation of lower peak
- Results suggest that we may expect the lower peak of the bimodal eigenvalue distribution in Fig. 2 to
continue to be better represented as the amount of scatter in the analyzer is increased.
- While the upper peak has not appeared for the types of scattering slab used in this paper, the results of
Sec. IV C 2 suggest that both peaks may grow as the amount of scatter is further increased.
## Possible reasons for lack of upper peak
- The relatively low number of eigenvalues near T=1 means that statistical convergence in this region
may not have been achieved.
- Using an incomplete set of channels when calculating the transmission matrix could cause this T=1
peak to fail to appear, though our simulations use the full set of propagating plane-wave modes.
## Quarter-circle distribution
- The quarter-circle distribution originated from random matrix theory.
- The (real-valued) singular value distributions of Re(T) and Im(T) are compared to the quarter-circle
distribution in Fig. 6.
- The decision to compare these matrices’ singular value distributions to the quarter-circle distribution is
not merely a practical one: it is actually the more correct choice of comparison in a rigorous sense.
## Empirical singular value distributions
- The matrix T already satisfies one of the quarter-circle distribution’s requirements.
- If the matrix entries are independently distributed as well, then the matrices Re(T)+Re(T)^T and
Im(T)+Im(T)^T would be expected to have eigenvalue distributions that follow the quarter-circle
distribution.
- The quarter-circle distribution has been found experimentally when studying the singular value
distribution of transmission matrices for disordered media.
## Applicability of the quarter-circle distribution
- A proof is included in this section that the matrices T and THT cannot simultaneously satisfy the
assumptions of the quarter-circle distribution.
- This is done by assuming the statistics of T and proving that the diagonal elements of THT are
distributed according to an Erlang distribution rather than the required exponential distribution.
## Erlang and χ2 distributions
- The Erlang distribution with rate parameter γ=1/2 and shape parameter k=n/2 is equivalent to a χ2
distribution.
- If the number of Gaussian random variables n is even, then their sum of squares follows an Erlang(
n/2,1/2) distribution.
## Statistics of THT
- The diagonal elements of THT follow a χ2 distribution, which is equivalent to an Erlang( N,1/2)
distribution.
- The statistics resulting from (13) hold a certain conceptual similarity with those mentioned in Sec. II.
- Zero-mean circular Gaussian field statistics result in intensity statistics that follow an exponential
distribution.
## Statistics of the quarter-circle distribution
- The matrix elements are not independent and do not approach independence for the slabs that were
simulated.
- The eigenvalues of the matrices Re(T) and Im(T) more closely approach a quarter-circle distribution as
the field more closely approaches zero-mean circular Gaussian statistics.
## Super-resolution far-field sensitivity
- The use of a randomly scattering analyzer can enhance subwavelength sensitivity to either a shifting
incident electric field or a changing field associated with a remote shifting object.
- Super-resolution sensitivity can be demonstrated by distinguishing between two features that are
separated in space by a subwavelength distance.
## Far-Field Sensing System
- The aperture construction has a total size of λ, making the far-field distance approximately 2λ away.
- The analyzer can be considered a far-field sensing system as it is located at this distance from the
aperture construction.
## Speckle Intensity Correlations
- The correlations plotted in Fig. 8 were calculated over the changing detected speckle intensity patterns
that resulted from translating the aperture system through a normally incident plane-wave background
field.
- Without a scattering analyzer, the one-aperture and two-aperture cases are difficult to distinguish.
- When a scattering analyzer is added, the decorrelation curves become more distinguishable,
demonstrating far-field super-resolution sensing.
## Using Standard Slab Types
- Using the same types of scattering analyzers that are parameterized in Table I (except for the highestscatter type), some expected patterns are noticeably absent.
- For the lowest-scatter and low-scatter analyzer types, an increase in scatter results in a faster
decorrelation as well as greater separation between the one-aperture and two-aperture curves.
- This increases sensitivity and reinforces prior results.
## Reduced Dielectric Constant
- Simulating analyzers with a lower dielectric constant, /epsilon1r=2, a more complete pattern emerges.
- By increasing the thickness of the scattering analyzer, the speckle decorrelates more quickly, resulting
in greater sensitivity.
- This suggests that increasing the thickness of a randomly scattering analyzer may be a preferable
method of increasing far-field subwavelength sensitivity.
## Object Function Comparison
- For the no-analyzer case, the two-aperture correlation decays slightly more quickly than the oneaperture correlation.
- The addition of the small PEC segment in the two-aperture case slightly narrows the far-field aperture
response.
- The decorrelation rate order appears to reverse when a scattering analyzer is added.
## Introduction
- The study focuses on speckle intensity correlations over position and the Fourier magnitude of the
incident field spectrum.
- The double aperture system has a narrower Fourier transform and slower decorrelation with use of the
analyzer.
## Access to Subwavelength Object Information
- The motion effect and changing fields sensed by the analyzer provide access to subwavelength object
information.
- Far-subwavelength object information can be encoded from the near field into the propagating
spectrum.
## Speckle Intensity Statistics and Scattering Regime
- The speckle intensity patterns attained in the study are inspected further.
- The introduction of a PEC film with a small aperture(s) has dramatically changed the field statistics.
- The results presented herein extend beyond previous work: super-resolution far-field sensitivity to a
subwavelength change is not limited to one particular type of field statistics.
## Proposed Explanation
- An explanation for the presented super-resolution results is proposed.
- Far-subwavelength object information can be encoded from the near field into the propagating
spectrum.
- Spatial sensitivity is enhanced with a heavily scattering analyzer.
## Randomly scattering analyzers
- The paper develops the theory of randomly scattering analyzers, which can be used for remote sensing
of all wave types that are coherent enough to produce an effect analogous to optical speckle.
- The presented random analyzer concept offers opportunities for sensing and microscopy where farsubwavelength spatial information is important.
## Understanding random matrix theory
- The eigenvalue distribution for the transmission matrix that represents a scattering analyzer provides
an explanation for why a more heavily scattering analyzer results in faster speckle intensity
decorrelation.
- The increase in sensitivity made possible by a randomly scattering analyzer is accompanied by a peak in
the distribution of eigenvalues of T.
## Super-resolution sensing capability
- The paper demonstrates a far-field super-resolution sensing capability.
- Measured intensity data through a random analyzer can be inverted to form a super-resolution image.
## Regulating statistical properties of a random medium
- The paper raises the prospect of regulating the statistical properties of a random medium to achieve
high spatial sensitivity to changes in the incident field.
- Designing a scattering analyzer using an asymmetric-transfer-function metasurface could result in
increased sensitivity compared to a random analyzer.
## Limitations of the technique
- Uniqueness of the data is a relevant concern for both sensitivity and imaging.
- The accuracy and precision of the measurements involved relate to the signal-to-noise ratio (SNR) of
the measurements and number of bits used by the sensor’s analog-to-digital converter (ADC).
## Potential subsequent steps
- Optimizing the spatial profile of the incident field could be combined with the analyzer concept
discussed here, possibly resulting in further improvements of estimation performance or greater
sensitivity.
- Estimation of the axial separation of two incoherent point sources using a mode sorter might be
reimagined in the context of a random analyzer.
## Relationship between structures and random medium
- Aperiodic structures have field-control properties that are dependent on the specific geometrical
features of the structure.
- There is a relationship between structures that may be designed for a specific task and the general
statistical character of a random medium acting as an analyzer for super-resolution spatial sensing.
## Conclusion
- The paper offers another dimension for metrology and other application domains.
- The presented random analyzer concept offers opportunities for sensing and microscopy where farsubwavelength spatial information is important.
- The remote sensing approach with a randomly scattering analyzer is applicable to all wave types for
which a heavily scattering medium exists and that are coherent enough to produce an effect analogous
to optical speckle.
## Distribution of T
- The lack of a peak in the distribution near T=1 is not associated with a lack of fit between the quartercircle distribution and the singular value distribution of either Re(T) or Im(T).
- Results show good agreement with the quarter-circle distribution when compared to appropriate fieldbased data sets.
## Acknowledgments
- This work was supported by the Air Force Office of Scientific Research under Grant No. FA9550-19-10067, and by the National Science Foundation (NSF) under Grants No. 1909660 and No. 21101633.
- The two anonymous referees who reviewed the paper provided insightful comments that were much
appreciated.
## Simulation Details
- The accuracy of any FEM simulation depends heavily on the size of the mesh elements.
- The mesh density was primarily controlled by reducing the maximum mesh element size, while
allowing the elements to be smaller than this by a factor of 10 as necessary.
- The necessary mesh densities were found to differ for the eigenvalue simulations of Sec. III and the
aperture simulations of Sec. V.
- The scattering properties of the simulated version of the analyzer are different from those of the
analyzer used in previous experiments.
- The incident plane wave was specified in the FEM software as a background field and the scattered
field was solved for.
- The spectrum of plane waves used for the transmission matrix simulations was chosen to be the full set
of propagating modes that are periodic in the geometry.
- For the aperture simulations, only the m=0 mode was used for the incident field, which corresponds to
the plane wave normally incident upon the scattering analyzer.
- The detector plane was positioned a distance of 4λ away from the scattering analyzer, along which all
the detector points were located.
## Convergence analysis
- Gradually increasing the distance between the medium-scatter slab and the detector plane showed a
change in mean CR of about 0.5% between detector distances of 4λ and 5λ.
- The statistics became closer to a zero-mean circular Gaussian at 4λ than at 1λ.
## Optical vortices
- Small regions of alternating positive and negative time-average Poynting vector were noticed in the
simulations.
- Their effect had significantly decayed by the chosen distance of 4λ.
## Transmission matrix simulations
- N points were uniformly distributed along the detector plane, corresponding to the N plane waves.
- The z component of the total electric field (Ez) was calculated at these points.
- N+1 uniformly distributed points were calculated along the detector plane to gain true periodicity.
- These N values were then passed through a discrete Fourier transform to calculate the entries of the
matrix T.
## Aperture simulations
- Detectors were uniformly distributed along the detection plane with a separation interval of λ/100.
- The speckle intensity pattern was calculated at the detector plane for each position of the aperture
system.
- These speckle patterns were pairwise correlated using the Pearson correlation coefficient.
- The average Pearson correlation coefficient for relative distance Δ1sx was calculated.
## Shift distance and random analyzer
- The aperture system was shifted in intervals of λ/10 parallel to the x-axis, up to maximum distances of
λ/2 in each direction.
- 1 ≤ n ≤ N are referred to as the 10 different instances of the random analyzer.
- The average Pearson correlation coefficient for relative distance Δ1sx was calculated using the Pearson
correlation coefficient between aperture positions sx1 and sx2 for the nth random analyzer.
## Symplectic character of transfer matrix
- The electromagnetic scattering matrix is reciprocal and unitary, resulting in SH= S^-1.
- Time-reversal invariance gives S=S/latticetop, where /latticetop denotes the (nonconjugate) transpose.
- The transmission matrix, T, was used in this paper.
- The eigenvalues of M^HM are real because the matrix product is real and symmetric.
## Conservation conditions
- Conservation of power (through the Poynting vector, or energy per unit time) is imposed.
- Conservation conditions lead to M^HΣ1M=Σ1.
- Σ1 is a matrix that represents conservation of current flux.
## Bimodal probability density function
- The bimodal probability density function is developed for completeness.
- The eigenvalues of M^HM belong to matrices.
- The transfer matrix is useful for cascading systems.
- The overall transfer matrix becomes simply a product of each subsystem transfer matrix.
## Symplectic nature of transfer matrices
- Real transfer matrices are symplectic, as are products.
- Mis symplectic under the more typically used definition, involving a skew-symmetric matrix.
- The symplectic nature of Mis proven by decomposing M into real submatrices to yield
M/Omega1M=Omega1, when combined with (B11) and (B12).
## Relating the scattering and transfer matrices
- Applying (B14) with (B7) and (B9) yields M-1 = [AH-CH, -BHDH].
- Using (B7), (B15), (B18), and (B21), it is found that MHM + (MHM)-1 = 2[AHA+CHC, 0; 0, BHB+DHD].
- Substituting (B30) and (B31) into (B22) [using inversion] gives 1/4[MHM + (MHM)-1 + 2I]^-1 = 1/4[THT,
0; 0, TTH].
## Transmission eigenvalues and conductance
- The conductance, g, is given by Tr(THT) (in units of e2/h, with a factor of two to account for spin).
- The mean conductances of each of the types of randomly scattering analyzer were calculated, and the
results are shown in Table V.
- These conductance values decrease as the amount of scatter in the analyzer increases, while the
number of eigenchannels remains constant (N=141).
## Bimodal distribution
- As the level of scatter increases, we approach the limit in which the bimodal distribution applies.
- The conductance can be written exactly in the limit as the sum of cosh2xn, which leads to a bimodal
density function for Tn.
## Transmission matrix and transfer matrix eigenvalues
- The transmission matrix M can be decomposed in an orthogonal basis as M=U√(I+λ)√(I+λ)√V.
- The diagonal matrix λ contains the unique eigenvalues of X, which can be used to calculate the
conductance.
## Uniform distribution for xn
- The Lyapunov behavior is established from the large length limit and can be represented as the
multiplication of M.
- The Lyapunov coefficients xn are uniformly distributed, which results in a uniform density function for
xn.
## Bimodal density function
- The probability density function for Tn can be developed into the bimodal density function.
- The bimodal density function is derived using a simplification of random variable notation and the
assumption that xn is uniformly distributed.
## Uniform Density Function
- The uniform density function over a continuous interval is given by pX(x) = 1/SX if x is in [0, SX], and 0
otherwise.
- A point X=x is related to Y=y by x = cosh^-1(sqrt(y)).
## Bimodal Eigenvalue Density Function
- The bimodal eigenvalue density function is obtained by forming pY(y) = Aδ1/y√(1-y), where δ is some
small positive value.
- To form a density function with a nonintegrable singularity at y=0, we have A^-1δ = integral from 0 to
δ1/y√(1-y) dy.
## Quarter-Circle Distribution
- The quarter-circle distribution originated during a derivation by Wigner of the eigenvalue distribution
of random-sign real symmetric matrices.
- It deals with real-valued symmetric matrices of dimension 2N+1, where N is very large.
- The diagonal elements are 0, and the nondiagonal elements vij = vji = ±v all have the same absolute
value v but random signs.
- The quarter-circle distribution is given by σ(x) = sqrt(8Nv^2 - x^2) / (4πNv^2) for -v√(8N) < x < v√(8N).
## Complex Matrices
- The quarter-circle distribution was further extended by Marčenko and Pastur in several ways, notably
allowing for complex-valued matrices.
- They derive an expression for the cumulative density function v(λ;BN(n),c), as well as its first derivative
with respect to λ, which is the corresponding probability density function.
- As N→∞, this eigenvalue density function converges to dv(λ;BN(n),c)/dλ = 4cτ^2 - λ^2 / (2πcτ^2 *
(1+λ+τ/τc) - 1)) for λ^2 <= 4cτ^2, where τ is any of the independent and identically distributed τi.
- In the limit c→∞, this becomes the same quarter-circle probability distribution that was arrived at by
Wigner.
## Normalization procedure in Sec. IV A
- The explicit dependence on N and m has been dropped through the normalization procedure.
- The required conditions for this distribution are met by independent and identically distributed
Gaussian entries.
## Alternative approach
- An alternative approach would have been to use Mar ˇcenko and Pastur’s work instead of Wigner’s.
- This would allow the complex-valued matrix T to be investigated directly, rather than the real-valued
Re(T) and Im(T).
Boundary condition for the optical force density
The paper discusses the derivation and implications of the boundary condition for the optical force
density in materials. It explores the relationship between the total force and the force density, taking
into account conservation principles and locally homogenized constitutive parameters. The paper
emphasizes the importance of understanding optical forces in various optomechanical phenomena and
applications, such as quantum cooling, molecular optomechanics, and biophysics. It also highlights the
need for experimental validation and further research in this area.
Certainly! The paper addresses several key issues related to the boundary condition for the optical force
density in materials.
Firstly, the paper emphasizes the importance of considering the force density as the fundamental way to
understand optical forces in materials [1]. The force density provides a spatially and temporally
dependent description of the forces, allowing for the integration and prediction of motion in threedimensional space. Conservation of momentum is derived from the force density, and continuity of the
normal component of the Abraham momentum density is required at material interfaces [1].
The paper also discusses the implications of the derived boundary condition for various optomechanical
phenomena and applications. It highlights the significance of understanding optical forces in fields such
as quantum cooling, molecular optomechanics, and biophysics [1]. The boundary condition provides a
framework for analyzing and interpreting experimental situations, and it is noted that new experiments
are needed to extract force (density) information near material surfaces [7].
Furthermore, the paper addresses the limitations and exceptions of the derived boundary condition. It is
mentioned that the maximum pressure predicted by the boundary condition applies in specific
situations, such as with a perfect mirror, but pressures outside of this range are possible depending on
factors like excitation, geometry, and time-history [3]. Systems with optical gain, for example, can have
negative pressure [3].
The paper also acknowledges the nonlocal-in-time contributions to the optical force density from both
the material and the structure. The temporal history of the system plays a role in determining the
mechanical response at any given time [4].
In conclusion, the paper highlights the need for further experimental validation and the development of
a generalized force theory that considers material and spatial dispersion. Such a model would have
broad applications in the physical sciences and related technologies [7].
## Introduction to Optical Forces
- The theory of electromagnetic forces lacks complete understanding of how photons interact
mechanically with materials.
- A force density theory based on the work of Einstein and Laub is used to develop boundary conditions
appropriate for material interfaces and optical frequencies.
- This leads to a theory for the total force and pressure that can be evaluated through various
experiments and applied to other models.
## Applications of Optical Forces
- Optical forces have become important in biology, physical chemistry, and condensed-matter physics.
- Optical rheology and mechanotransduction in cells have enabled new experimental regimes.
- Optomechanics has led to surprising findings in classical statistical mechanics, including anomalous
attraction, oscillatory colloidal interactions, and hydrodynamic fluctuations.
- Optical traps have been studied extensively for use in achieving Bose-Einstein condensates and in
regards to photon momentum exchange.
- Nano-optomechanical actuators have been demonstrated that have the potential to impact optical
signal processing.
- Other opportunities in quantum information processing, thermal and humidity sensing, optical logic
gates, channel routing or switching, dispersion compensation, and tunable lasers have also been
considered.
- All-optical means of communication between computers and in radio frequency photonics have been
explored.
## Optical Force Density and Model
- To understand how light imparts a force throughout condensed matter, one must build a theory for
the force density.
- The force density theory considered here stems from work done by Einstein and Laub, yielding results
of note for the conservation of momentum in physical materials.
- The goal of the present work is to evaluate the requirement of the optical force density at material
interfaces, in order to correctly connect the force density internal to a material to the external force, for
example, to obtain the pressure.
## Boundary Condition for the Optical Force Density
- The boundary condition associated with the optical force density is developed and investigated using
an expression stemming from the work of Einstein and Laub, and in conjunction with Maxwell’s
equations to describe the electromagnetic fields.
- A constraint is formed that allows a unique relationship between the total force and the force density,
one that is achieved by virtue of the conservation principles for physical materials and described by
locally homogenized constitutive parameters.
- The mathematical steps presented form a basis for modeling various optomechanical phenomena,
including optical forces in and on solid-state systems such as membranes, beams, cantilevers, and
waveguides, and can be interpreted in terms of a suite of related theoretical work.
## Experimental Studies
- Further insight can be garnered from new experimental studies, as summarized.
- Section IV considers mathematical and physical interpretations in relation to experiments, including
suggestions for studies to evaluate the theory, as well as broader impacts in current application spaces.
## Relevance of the Force Density Boundary Condition
- This specification of the force density boundary condition is relevant for basic scientific fields including
those involved with various quantum cooling issues, molecular optomechanics, photochemistry, and
biophysics (including mechanotransduction).
- The technologies impacted encompass integrated optomechanics (silicon photonics, where new optical
device concepts can be enabled), communication systems (in which optical forces could supplant
electronic switching), remote control and actuation, propulsion, sensing, and navigation.
## Open Questions Related to Optical Force Density
- More than one century after the first measurements of optical force, there remains such uncertainty
with regard to force densities in condensed matter.
- To resolve open questions related to optical force density, a combined basic theoretical and
experimental effort is needed.
- The key interface problem treated here relates to this goal.
## Mathematical Treatment
- Section III provides background on the force density model, and the specific boundary issues
developed are introduced.
- The mathematical treatment in Section III is the key contribution of this work.
## Introduction
- Dielectric beads rely on a gradient force involving the spatial derivative of the electric field.
- The dipole force density is important in such inhomogeneous material systems.
## Force Density Boundary Condition
- Maxwell's equations are written with all source terms on the right-hand side.
- The Abraham momentum density is given by g=1/c^2 E×H.
- The Einstein-Laub momentum flow or stress tensor is defined as T=1/2(εE·E+μH·H)I−DE−BH.
- Equation (17) provides a basis to consider arbitrary material responses.
- For source-free, nonmagnetic materials, (17) becomes ∂g/∂t=−∇·T−(∂P/∂t)×μ0H+(P·∇)E.
## Electromagnetic Fields and Related Mathematics
- Maxwell's equations are written with all source terms on the right-hand side.
- The cross product of εE with (1) and μ0H with (2), and adding the resulting equations, gives (6).
- The divergence of the tensor in the (x1,x2,x3) coordinate system is given by (8).
- The dyadic product of two vectors is defined according to (ab)ij=aibj, giving, for example, EE as in (9).
## Boundary Conditions
- Maxwell's curl equations applied to boundaries with sufficient smoothness and for physical materials.
- Loss and hence free charge motion is incorporated into the temporal Fourier form for the polarization.
- For isotropic materials, the complex electric susceptibility χE and dielectric constant ε1 are used.
- Equation (18) is obtained for source-free, nonmagnetic materials.
## Tangential field boundary conditions
- The tangential field boundary conditions are given by continuity of tangential magnetic and electric
fields.
- These conditions combined with the momentum density yield a momentum density boundary
condition.
## Preservation of normal component of momentum density
- Equation (21) implies that the normal component of the Abraham momentum density is preserved
across an interface.
- This is due to the field boundary conditions and indicates that the momentum density in media with χE
other than zero has the same form as that in vacuum.
## Preservation of normal components of Poynting vector
- The normal components of the Poynting vector (S=E×H) are also preserved across interfaces.
- This is a statement of energy conservation based on Poynting’s theorem.
## Simplification of optomechanics problem
- Assuming that the condensed matter experiencing an optical force is fixed in position for a time that is
large relative to the temporal period of the optical wave, the complex spatiotemporal optomechanics
problem can be simplified.
- This is representative of many experimental situations and can be evaluated using computational
electromagnetics.
## Evaluation of boundary-condition requirements
- Equations (25) and (26) are key to evaluating boundary-condition requirements for the kinetic force
density.
- Selecting the normal components from (26) yields an equation that is correctly interpreted in the
temporal Fourier domain and with spectral superposition.
## Ascription of kinetic force density
- At the interface between two materials, there is in general an electric-field component normal to the
interface.
- In the mathematical (and nonphysical) case of a perfect electric conductor, an unlimited charge can
respond instantaneously, and the surface charge density would experience a force just as Lorentz
described.
## Local force on a charged surface
- In situations where there is a net charge on a local solid-state material, the charge experiences a force.
- This force can be described as a local force on a charged surface.
## Optical frequencies and neutral materials
- At optical frequencies, moving a neutral material with a kinetic force requires collective motion of
electric dipoles.
- Free charge motion can be neglected in the boundary representation for optical forces in neutral
materials.
- The force density at the interface cannot be infinite physically in such neutral materials.
## Electromagnetic momentum density
- The electromagnetic momentum density in material has a long history of debate.
- Photon momentum must flow across an interface, and the relevant components of the momentum
flow tensor must be continuous across the interface.
- The kinetic force density associated with collective material motion can be described using the Cauchy
principal value and without boundary Dirac delta terms.
## Conservation of momentum
- All relevant boundary conditions are imposed, and momentum flow is conserved.
- The total kinetic force can be described using a volume integral in the Cauchy principal value sense.
- Both the temporal and spatial integration requirements are addressed.
## Experimental studies
- Experimental studies with membrane deflection support the existence of enhanced optical pressure.
- New experiments can explore the forces near interfaces and at the nanometer length scale.
- The impact on several current research trends is outlined.
## Flexural modes in SiN membranes
- SiN membranes used in optomechanics experiments have lower-order resonances below the MHz
range.
- A thin-plate model suggests that the local linear velocity is of the order of 10^3 m/s.
## Neglecting material motion in optomechanical analysis
- Material motion might be neglected in the optomechanical analysis for solid-state systems over optical
periods and cavity excitation times in plasmonics.
- A temporal average over the carrier period presents the modulated light description and the plausible
time frame for a mechanical response.
## Time-averaging in force density
- Modulated light and a local average over the optical (carrier) having period t0 can be described to form
averages of the force density and total force or pressure.
- This produces a (modulated-light) time variation in the force quantities that is commensurate with
mechanical motion timescales.
## Viewing optical forces through forced density
- Optical forces in materials should be viewed through the spatially and temporally dependent force
density.
- This forms the basis to obtain components of force and torque through integration and to predict
motion in 3D space.
## Conservation of momentum
- All materials must have continuity of the normal component of the Abraham momentum density.
- The constraints involving ∂g/∂t and Tij must hold at interfaces, as well as along any contour within a
locally homogeneous domain.
- The total kinetic force, that which can move a structure in a manner measured by spatial motion or
matter, would appear to be dictated by (33).
## External momentum flow tensor
- Equations (32) and (33) are applied within the spatial support of the medium.
- The result from the integration of the Dirac delta at the surface is added to the stress tensor
integration to obtain the same result as from the spatial integration within the material.
- This produces a consistent internal and external description of the kinetic force and allows correct
interpretation of the time-dependent momentum flow into the material.
## Nonlocal-in-time fields and impact on the force density
- The fields in all physical materials are a result of current and all earlier excitations in time, making the
description non-local in time.
- The nonlocal-in-time concept for optical forces is important and essential in the correct theoretical
description, and it is incorporated into the theory in (32) and (33).
## Materials with gain
- Materials with gain (optical activity) can impact the energy exchange with the outside world and also
the force imparted.
- Optical cavities in various forms provide energy storage and this impacts the force.
## Introduction
- Negative pressure and time history of the system are important factors in understanding momentum
flow into a volume.
- The force density provides a fundamental path to kinetic information.
## Momentum Flow into a Volume
- Jackson's presentation of momentum flow into a volume and conservation conditions is widely
disseminated.
- The conservation condition relates to the total time rate of change of mechanical plus field momenta
and does not in general allow the two to be distinguished.
- There is no information about the spatiotemporal distribution of the forces in the volume.
- The relevance of the boundary description has been presented.
## Boundary Conditions
- Boundary conditions that must hold at interfaces are considered.
- Continuity of momentum flow is required in a vacuum environment.
- Additional considerations are needed to preserve the momentum flow from vacuum into and out of
the material.
- The electromagnetic system described may be open, so other forces can be involved.
## Coupled System Forces
- Coupled system forces need to be considered to solve the kinematic problem in relation to
experiments.
- The right-hand side of the conservation condition and photon momenta through an external boundary
does not provide the fundamental conservation condition related to motion of media.
- With pulsed light, the right-hand side of the conservation condition could be zero and only the field
momentum gives rise to nonzero mechanical momentum.
- Care is needed in applying the time domain form of the conservation condition to infer the kinetic
force.
## Sinusoidal Steady State
- The sinusoidal steady-state case having a single circular frequency is mathematically and
computationally convenient.
- Such a model could also be of practical importance because relatively coherent laser light or lowfrequency modulated.
## Introduction
- Excitation to a quasimonochromatic case may need only be long compared with a measure such as the
average photon lifetime in the material.
- The system modeled must be linear and time-invariant.
## Sinusoidal steady-state kinetic force density
- The time-average sinusoidal steady-state kinetic force density can be written as a function of
polarization, electric field, and magnetic field phasors.
- The average total force becomes the integral of the force density over some volume.
- The physical situation described is one where the entire constellation of materials has been invariant
for an infinite time.
## Boundary condition for momentum flow
- To conform to a steady-state field solution, the media must be immobilized.
- The boundary condition for the photon flow must be imposed on the right of the equation.
- Without including contributions from [P(r,t)·∇]E(r,t), the physical material boundary condition is
violated.
## Modeling the general time-dependent optical force
- Optomechanics involves the coupled electromagnetic and mechanical problems where momentum is
transferred from fields to matter.
- The Yee algorithm utilizes displaced Cartesian grids for formation of the curl operators and finite
difference representations.
- The optomechanical problem can be treated in a manner similar to the FDTD method.
- Extensions of the FDTD approach may be a simple way to implement an algorithm for the general
coupled optical and mechanical problem.
## Maxwell's description of force and energy density
- Maxwell's description of force on a planar surface is related to electromagnetic energy, leading to the
'stress of radiation'
- The underlying meaning of this equation equates an electromagnetic field result to the negative of the
kinetic force density
## The Maxwell-Bartoli pressure
- The Maxwell-Bartoli pressure was presented by Nichols and Hull and Lebedev
- It is widely used for understanding the pressure on opaque materials
- The form of the equation can be inferred from situations that allow it
## Deriving equation (44)
- Equation (44) can be derived for the sinusoidal steady state situation
- It involves writing the field as a superposition of incident and reflected plane waves
- The frequency domain forms of (7) or (14) or the right-hand side of (43) provide for this result
## Pressure outside of [0, 2/Si]c is possible
- Depending on factors involving excitation, geometry, and time-history, pressures outside of [0, 2/Si]c
are in principle possible
- Systems offering optical gain can have negative pressure
- Asymmetric plasmonic cavities were found numerically to have a pressure greater than 2/Si/c
- There is a proposal for a system with surface wave modes induced on the back that may yield a
negative pressure
## Conditions for exceeding 2/Si/c
- With imposition of the momentum flow boundary conditions implied by (25) and (26), it becomes
possible to exceed 2/Si/c
- This can be established from the right-hand side of (43) with the enforcement of momentum flow
continuity across boundaries
- This frequency domain revision to momentum flow is legitimate because the material is fixed in
position for infinite time and the resulting fields, force density, force and momentum must satisfy these
constraints
## Experimental studies
- Additional experimental studies are important to evaluate (32) for materials at optical frequencies
- An optomechanics experimental effort involved a SiN membrane with patterned Au that supported a
surface-plasmon mode
- Results from those experiments were found to be consistent with an enhanced optical pressure, one
that exceeded that with the same incident-field power density normally incident on a perfect mirror
- New experiments are needed to allow evaluation of the force density with optical electric fields normal
to interfaces
## Force density and pressure calculations
- From an electromagnetic field basis, the force density and pressure calculations applied within a
resonant plasmonic cavity and yielding enhancement can satisfy all physical requirements
- This also allows for the possibility of pulling structures with light in a manner that is fundamentally
different from optical tweezing of beads in a trap
- Evidence of pulling would provide further support for (32)
## Conclusion
- Equation (44) is a statement of momentum flow through a surface and for a single-plane-wave
problem with a local material response
- New experiments are needed to evaluate the force density with optical electric fields normal to
interfaces
- Further studies are needed to validate the force density and pressure calculations
## Challenges in optical force experiments
- Obtaining precise displacement precision is straightforward, but experiments need to provide a
uniquely interpretable force density result.
- Beam characterization, adequate measurement diversity, and adjustment of key parameters are
necessary.
- Parametrized models may need to be utilized that require careful consideration.
## Applications of optical force density theory
- Improved understanding of the theory for the optical force density in condensed matter will have
immediate ramifications in several developing application domains.
- Patterned silicon nitride (SiN) membranes, tethered trampoline geometries, and photonic crystals are
some examples.
- Optical forces in nanostructured material may aid photomotility.
## Optical force and enhanced pressure
- The optical force density in (32) and total force in (33) result in an enhanced optical pressure that
exceeds the Maxwell-Bartoli result in (44).
- This effect requires an excitation time that is long relative to the cavity lifetime and a system force that
resists motion over timescales commensurate with the optical period and cavity lifetime.
- Interpretation of a sinusoidal steady-state force model to an experimental situation needs to be
considered in relation to the specific situation.
## Maximum momentum exchange
- Maximum momentum exchange with ¯ hk0 per photon leads to 2/⟨Si⟩/c being the maximum force
density for opaque materials.
- This is true only in specific situations and not that considered in Ref. [90].
## Internal and external descriptions
- The internal and external descriptions are coupled.
- The exterior mathematical picture comes from various steps stemming from the force density in
material and so cannot be independent of the result inside the material.
## Physical picture
- The assumptions in the model and the relationship to an experimental situation is misstated in Ref.
[91].
- A distinction is made between forces on part of the object and the total force.
- The object moves most fundamentally due to the spatiotemporal mechanical force density.
## Mathematics
- Using the external, free-space stress tensor to circumvent a nonphysical Dirac δ function at surfaces is
fundamentally wrong.
- There remains a Dirac delta term at dielectric interfaces.
- The total force results in Ref. [91] are effectively determined using one approach, that involving a freespace stress tensor in the frequency domain.
## Pressure enhancement
- The theoretical development of the force density boundary condition proves that pressure
enhancement is possible.
- It would seem prudent to emphasize force density in a path forward and not rely on inferences from a
free space stress tensor.
## Fundamental interaction of light with condensed matter
- This work probes the fundamental way in which light interacts with condensed matter and transfers
momentum from the photon to the material.
- A boundary condition relevant for the force density in materials and inhomogeneous condensed
matter that provides a basis for the unique interpretation of total force is derived.
## Consideration of interface surface charge densities in force calculations
- The interface surface charge densities should be considered in force calculations for statics and even
cryogenic temperatures.
- New experiments are needed to extract force (density) information near to the surface in solid-state
materials.
## Boundary conditions for momentum flow across boundaries
- The theory developed here describes the boundary condition that allows momentum flow, Poynting
vector, and field boundary conditions to be simultaneously satisfied.
- The force density boundary condition is shown to be an essential part of the correct description of
interfaces.
## Implication of the model in relation to motion
- Frequency domain models describing sinusoidal steady state preclude motion.
- The complex and coupled electromagnetic and mechanical problem is greatly simplified through the
assumption that measurable motion of macroscopic objects cannot occur during the optical period.
## Nonlocal-in-time contributions to the optical force density
- The material and the structure both provide nonlocal-in-time contributions to the optical force density.
- The temporal history dictates the mechanical response at any given time.
## Acknowledgments
- The research was funded by The Air Force Office of Scientific Research and the National Science
Foundation.
- The author thanks the two anonymous referees and the PRB editors for helping improve this paper.
## Development of the photon momentum
- The peak value of the instantaneous momentum density is given by N¯hk0.
- This gives the vacuum photon momentum as p0=¯hk0, in accordance with quantum theory.
## References
- A list of references is provided for further reading.
## Introduction
- The article discusses various studies related to physics and optics.
- The studies include research on optical forces, optomechanics, and cavity optomechanics.
## Optical Forces
- Studies have been conducted on optical forces since the 1970s.
- Recent studies have focused on the boundary conditions for optical forces and the electrodynamics of
moving media.
## Optomechanics
- Optomechanics is the study of the interaction between light and mechanical motion.
- Recent studies have focused on the use of optomechanics in quantum information processing and the
development of optomechanical devices.
## Cavity Optomechanics
- Cavity optomechanics is the study of the interaction between light and mechanical motion in a cavity.
- Recent studies have focused on the use of cavity optomechanics in the development of quantum
technologies and the study of fundamental physics.
## Notable Studies
- Studies by Crocker et al. and Verma et al. focused on the behavior of colloidal particles in optical traps.
- Studies by Yodh et al. and Wang et al. focused on the thermodynamics of small systems.
- Studies by Davis et al., Zwierlein et al., and Campbell et al. focused on the behavior of ultracold atoms.
- Studies by Meekhof et al. and Brewer et al. focused on the manipulation of ions using lasers.
- Studies by Kippenberg and Vahala, Aspelmeyer et al., and Peano et al. focused on the use of
optomechanics in the development of quantum technologies.
- Studies by Fang et al. and Malz et al. focused on the use of cavity optomechanics in the development
of optomechanical devices.
- Studies by Deng et al. and Tao et al. focused on the use of optomechanics in sensing applications.
- Studies by Rosenberg et al., Nikolova et al., and Lin et al. focused on the development of photonic
devices using optomechanics.
## Conclusion
- The studies discussed in the article demonstrate the wide range of applications for optomechanics and
cavity optomechanics.
- These studies have the potential to impact fields such as quantum information processing, sensing, and
fundamental physics.